A Finite Element Model for Direction-Dependent Mechanical

Home , Transverse isotropy

D. Carnelli Department of Structural Engineering, A Finite Element Model for Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, Direction-Dependent Mechanical 20133 Italy; Department of Materials Science and Response to Nanoindentation of Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 Cortical Bone Allowing for Anisotropic Post-Yield Behavior D. Gastaldi V. Sassi of the Tissue

R. Contro A finite element model was developed for numerical simulations of nanoindentation tests on cortical bone. The model allows for anisotropic elastic and post-yield behavior of the Department of Structural Engineering, tissue. The material model for the post-yield behavior was obtained through a suitable Laboratory of Biological Structure Mechanics linear transformation of the stress tensor components to define the properties of the real (LaBS), anisotropic material in terms of a fictitious isotropic solid. A tension-compression yield Politecnico di Milano, stress mismatch and a direction-dependent yield stress are allowed for. The constitutive 20133 Italy parameters are determined on the basis of literature experimental data. Indentation experiments along the axial (the longitudinal direction of long bones) and transverse directions have been simulated with the purpose to calculate the indentation moduli and the C. Ortiz tissue hardness in both the indentation directions. The results have shown that the trans- Department of Materials Science and verse to axial mismatch of indentation moduli was correctly simulated regardless of the Engineering, constitutive parameters used to describe the post-yield behavior. The axial to transverse Massachusetts Institute of Technology, hardness mismatch observed in experimental studies (see, for example, Rho et al. [1999, Cambridge, MA 02139 “Elastic Properties of Microstructural Components of Human Bone Tissue as Measured by Nanoindentation,” J. Biomed. Mater. Res., 45, pp. 48–54] for results on human tibial P. Vena cortical bone) can be correctly simulated through an anisotropic yield constitutive model. Department of Structural Engineering, Furthermore, previous experimental results have shown that cortical bone tissue subject Laboratory of Biological Structure Mechanics to nanoindentation does not exhibit piling-up. The numerical model presented in this (LaBS), paper shows that the probe tip-tissue friction and the post-yield deformation modes play Politecnico di Milano, a relevant role in this respect; in particular, a small dilatation angle, ruling the volumet- 20133 Italy; ric inelastic strain, is required to approach the experimental findings. IRCCS, ͓DOI: 10.1115/1.4001358͔ Istituto Ortopedico Galeazzi, Milano, 20161 Italy Keywords: cortical bone, indentation, finite element, anisotropic yield function e-mail: [email protected]

1 Introduction hence, parallel fibered bone exhibits a transverse ͑normal to fiber axis͒/axial ͑parallel to fiber axis͒ elastic modulus ratio Ϸ0.42 ͓11͔. The technique of nanoindentation allows for the mechanical ͑ characterization of bone tissue at small length scales, thereby pro- Osteonal bone is less anisotropic, exhibiting a transverse r ͒ ͑ viding a pathway to explore the relationship between mechanical =normal to Haversian canal long bone axis /axial z=parallel to ͒ Ϸ properties and fundamental structural constituents ͓1–6͔. It also Haversian canal long bone axis /elastic modulus ratio 0.6 ͓ ͔ holds potential for use as a diagnostic tool ͓7͔. Traditional meth- 12,13 . The latter observation can be attributed to the fact that in ods of estimating material properties from indentation data in- osteonal bone, mineralized collagen fibrils bundle into layers with clude the Oliver–Pharr contact mechanical analytical formulation varying directions to form a plywood-like structure unit, as well ͓8͔, which defines an indentation modulus that, in the case of an as rotate around their axes within these layers ͓10,14͔. Cortical isotropic, elastic half-space continuum, can be explicitly related to bone is also known to exhibit a tension/compression yield asym- the elastic constants. Since bone is a hierarchical ͑multiscale͒ ma- metry ͓15͔, which is associated to a pressure-dependent mechani- terial, homogenized responses are achieved at sufficiently large cal behavior ͓16–18͔. length scales while the mechanical properties of distinct micro- Analysis of nanoindentation data using more refined constitu- and nanostructural features can be probed at smaller length scales. tive laws and finite element analyses ͑FEA͒ can provide deeper In addition to possessing a hierarchical structure, bone is also insights into these unique structural and mechanical features of anisotropic ͓9͔. Mineralized collagen fibrils, the fundamental the bone tissue ͓17–22͔. Previously reported literature in this area building blocks of bone, are structurally anisotropic ͓10͔, and includes the use of elastic anisotropy, plastic pressure-independent anisotropy, and strain hardening ͓20͔ and, separately, a nanogranular pressure-dependent yield models, such as the Drucker–Prager Contributed by the Bioengineering Division of ASME for publication in the JOUR- ͓17͔ or the Mohr–Coulomb models ͓18͔, capturing the known NAL OF BIOMECHANICAL ENGINEERING. Manuscript received July 20, 2009; final manu- ͓ ͔ script received February 18, 2010; accepted manuscript posted March 1, 2010; pub- tension/compression yield stress mismatch. Another study 21 , lished online June 18, 2010. Assoc. Editor: Ellen M. Arruda. utilized a viscoelastic-plastic, isotropic, constitutive model based

Downloaded 02 Jul 2010 to 18.19.0.100. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm on a spring-dashpot approach to viscoelasticity and on the transformation of equations for isotropic linear elasticity ͓24͔. Us- Ramberg–Osgood relationship for the isotropic post-yield behav- ing indicial notation, the stress-strain relationship for a linear elas- ior. While these works have provided significant new insights into tic, isotropic material in the small deformation regime is bone indentation behavior, accurate predictions of direction- ␴ = ␭⑀ ␦ +2␮e ͑1͒ dependent indentation moduli ͓23,1͔, direction-dependent hard- ij kk ij ij ness ͓1͔, and negligible pile-up ͓1,17͔ simultaneously still remain and the relevant stiffness tensor is a challenge. C = ␭␦ ␦ +2␮I ͑2͒ In this paper, we approach this problem by formulating a FEA ijkl ij kl ijkl model which captures elastic and post-elastic anisotropy, as well where ␴ij is the Cauchy stress tensor, eij is the deviatoric compo- as pressure-dependent yield ͑tension/compression asymmetry͒ of nent of the deformation tensor, ⑀kk is the specific volumetric de- cortical bone tissue. The elastic behavior of the tissue is modeled formation ͑summation over index k is assumed͒, ␭ and ␮ are the through the transversely isotropic elastic tensor ͓24͔ and the post- Lamè constants, ␦ij is the Kronecker symbol, and Iijkl is the elastic behavior is introduced via a yield surface and a flow rule fourth-order unit tensor. obtained by a suitable stress transformation of the Drucker–Prager 1 ͓␦ ␦ ␦ ␦ ͔͑͒ surface by means of a stress transformation rule to account for Iijkl = ik jl + il jk 3 material anisotropy. A tension/compression yield stress mismatch, 2 typical of the cortical bone tissue observed at the macroscopic In order to transform the above equations into a set of equations ͓ ͔ scale 15,16 , is accounted for. Parametric studies on the degree of for anisotropic materials, three perpendicular unit vectors g1, g2, tissue anisotropy, the friction between the indenter and tissue, and and g3 are introduced having the physical meaning of principal the amount of volumetric inelastic strain governed by the dilata- material directions; furthermore, the second-order tensor Gij is tion angle in a nonassociative plastic flow rule were carried out. introduced as follows: Load versus penetration depth was predicted and the Oliver–Pharr 1 1 2 2 3 3 ͑ ͒ analysis was used to estimate indentation properties. The Gij = p1gi gj + p2gi gj + p3gi gj 4 ͒ ء / ء ͑ transverse/axial indentation moduli ET EA , the transverse/axial where p , p , and p represent the weights of the three directions ͑ / ͒ 1 2 3 hardness values HT HA , as well as the pile-up, are predicted and in terms of elastic properties. We simulate an anisotropic material compared with experimental data on cortical bone reported in lit- by substituting the second-order identity tensor ␦ij with a new erature ͓1,4–6͔. second-order tensor IЈ = p ␦ + G ͑5͒ 2 Methods ij 0 ij ij so the anisotropic elastic constitutive law now reads 2.1 Elastic Anisotropic Model for Cortical Bone. Cortical ␭ Ј Ј ␮͓ Ј Ј Ј Ј ͔͑͒ bone is modeled as a homogeneous continuum. A transversely Cijkl = IijIkl + IikIjl + IilIjk 6 isotropic elastic model is used and obtained through a suitable In matrix notation it reads

␮ 2 ␭ 2 ␭ ␭ 2 r1 + r1 r1r2 r1r3 0.0 0.0 0.0 ␭ ␮ 2 ␭ 2 ␭ r1r2 2 r2 + r2 r2r3 0.0 0.0 0.0 ␭r r ␭r r 2␮r 2 + ␭r 2 0. 0.0 0.0 C = 1 3 2 3 3 3 ͑7͒ ␮ ΄ 0.0 0.0 0.0 r1r2 0.0 0.0 ΅ ␮ 0.0 0.0 0.0 0.0 r1r3 0.0 ␮ 0.0 0.0 0.0 0.0 0.0 r2r3

with conditions are satisfied is to define the properties of the real an- ͑ ͒ isotropic material in terms of a fictitious isotropic solid. This is ri = p0 + pi 8 achieved by relating the stress between the real and fictitious ͚3 spaces using a linear tensor transformation ͓26͔. The constraint i=0pi =1 is also introduced in order to obtain a set In this work, an extension of the Drucker–Prager plasticity iso- of independent constitutive parameters. The value p0 =1 has been tropic model ͓27͔ to the anisotropic case has been formulated by assumed. For transversely isotropic simulations, p1 =pA and p2 defining a suitable stress tensor projection into the anisotropic =p3 =pT, where the subscripts A and T refer to the axial and transverse directions, respectively. In the finite deformation regime, the space allowing for material principal directions g. The proposed above expounded elastic stress-strain relationships are used in in- modified Drucker–Prager model is able to account for the tension/ compression yield stress mismatch exhibited by the cortical bone cremental form ⌬␴ij=Cijhk⌬⑀hk with suitable rotations of the prin- tissue under macroscopic tension/compression mechanical loading cipal material directions gi during the deformation process. ͓15,28͔. 2.2 Yield Locus for Anisotropic Tissue. The post-elastic me- The plasticity domain for an isotropic material can be written in chanical behavior of the bone tissue was modeled using an elastic- the general form as plastic constitutive law with anisotropic yield locus and a nonassociative plastic flow rule. Traditional procedures for deriving the constitutive equations for anisotropic elastoplastic materials are ␾͑␴ ͒ =0 ͑9͒ based on the description of appropriate yield and potential func- ij tions in terms of the characteristic material properties and principal directions. The satisfaction of invariance conditions in these The second-order tensor M2 is introduced, the components of cases can be difficult ͓25͔. A procedure to guarantee that these which depends on the principal material directions g as follows:

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Downloaded 02 Jul 2010 to 18.19.0.100. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm 2 ␦ 1 1 2 2 3 3 ͑ ͒ say no strain hardening has been introduced. Mij = q0 ij + q1gi gj + q2gi gj + q3gi gj 10 The flow rule, which determines the irreversible strain ͑“plas- in which a proper weight is assigned to each material direction tic” strain͒ within the tissue is as follows: ء ء .through the parameters qi ␴ץ ͒ ␴͑ ץ ء ␴ p Q pq A new stress tensor ij projected into the anisotropic space is d⑀ = d␭ ͑16͒ ␴ץ ء␴ץ ij introduced pq ij ء .␴ ͑ ͒ in which the plastic potential Q͑␴ ͒ has been introduced 4 ء␴ ij = Mijkl kl 11 ͒ ͑ ء␴ ء ␣ ͱ ͒ء␴͑ in which the fourth-order tensor M4 is defined as Q =3 3 defp + ¯ 17 where ␣ is the dilatation angle. In the case in which ␣ =␣, the 1 def def 4 ͓ 2 2 2 2 ͔͑͒ Mijkl = MikM jl + MilM jk 12 associative plastic flow rule is obtained. 2 In the above flow rule, d␭ is a scalar parameter that can be The new yield locus for the anisotropic model is therefore ex- obtained by solving the nonlinear set of equations ͑Eq. ͑16͒͒, the p pressed in a general form as stress-strain incremental relationship d␴=C͑d⑀−d⑀ ͒ and the consistency condition d␾=0. These equations are solved by ͒ ͑ ͒ ء␾͑␴ ij =0 13 means of the iterative Newton–Raphson procedure applied at each The yield locus of the modified Drucker–Prager plasticity model integration point and at each time increment during the finite el- for the anisotropic bone tissue is as follows: ement simulation. To this purpose, the standard computational procedures for elastic-plastic finite element models have been ͒ ͑ ␴eq ء␴ ء ␣ ͱ ͒ ء␾͑␴ ij =3 3 p + ¯ − ¯ =0 14 adopted in which consistent tangent stiffness matrix has been computed to solve for the displacement vector ͑see, for example, ء␴ ء in which the projected stress components p and ¯ are defined as Ref. ͓29͔͒. 1 3 In the specific case, a transversely isotropic elastic-plastic be- ͒ ͑ ء ء ͱ ء␴ ء␴ ء p = ii; ¯ = SijSij 15 havior has been used in which the plane of isotropy of the tissue is 3 2 perpendicular to the axial direction for long bones by making the :following assumptions ء ␴eq ␣ and and ¯ are the material parameters; whereas Sij is the ͒ ͑ ء␴ deviatoric component of the Cauchy stress tensor ij. The param- q1 = qA; q2 = q3 = qT 18 ␣ are ءand ¯␴ ءeter rules the tension/compression yield stress ratio; whereas, Under these assumptions the stress components p the parameter ¯␴eq depends on the tensile or compressive yield ͑ ͒ 1 1 ␴ ͑q + q ͒2 + ͑␴ + ␴ ͒͑q + q ͒2 ͑19͒ = ءstress or elastic limit of the tissue. For sake of simplicity, a p perfect elastic-plastic material model has been assumed; that is to 3 1 0 A 3 2 3 0 T

͒͑͒ ͱ␴2͑ ͒4 ͑␴2 ␴2 ␴ ␴ ͒͑ ͒4 ͑␴ ␴ ␴ ␴ ͒͑ 4 ء␴ ¯ = 1 q0 + qA + 2 + 3 − 2 3 q0 + qT − 1 2 + 1 3 q123 20

͑ ͒ in which q0 =1; q0 + qA +2qT =1 26 4 4 ͑ ͒ 3 q123 = q0 + 2qA +2qT q0+ are adopted, thus only one of the q material parameters results to ͑ 2 2 ͒ 2 ͑ 2 2 ͒ 2 2 ͑ ͒ i + qA + qT +4qAqT q0 + 2qAqT + qAqT q0 + qAqT 21 be independent. The tensile and compressive yield stress along the axial direction 2.3 Finite Element Model of Indentation Tests. The virtual ͑␴t ␴c ͒ ␴ ␴ A , A can be obtained by setting 2 = 3 =0. indenter employed was axisymmetric, conical with an angle of ¯␴eq 1 ¯␴eq 1 70.3 deg and the same area-to-depth ratio as a three sided Berk- ␴t = ; ␴c = ͑22͒ ovich pyramid. The indenter was modeled as perfectly rigid since A ͱ ͑ ͒2 A ͱ ͑ ͒2 1+ 3␣ q0 + qA −1+ 3␣ q0 + qA it is composed of diamond, which is much stiffer than bone ͑ Ϸ ͒ whereas the tensile and compressive yield stress along the trans- Ediamond=1141 GPa; Ebone 20 GPa . The indentations along ͑␴t ␴c ͒ ␴ ␴ the axial, as well as transverse directions, were simulated. verse direction , can be obtained by setting 1 = 3 =0 T T Given the transversely isotropic constitutive behavior of corti- ¯␴eq 1 ¯␴eq 1 cal bone, an axisymmetric model was employed for indentation ␴t = ; ␴c = ͑23͒ T ͱ ͑ ͒2 T ͱ ͑ ͒2 along the axial direction. For indentations along the transverse 1+ 3␣ q + qT −1+ 3␣ q + qT 0 0 direction, a three-dimensional model was created. In the latter The tensile to compressive yield stress ratio turns out to be inde- case, only a quarter of the bone tissue/indenter system was mod- pendent from the material direction and ruled by the material pa- eled by exploiting the symmetry of the material at 90 deg and the rameter ␣ only. axisymmetry of the conical indenter. In both models the bone ␮ t ͱ sample is represented as cylinder of 100 m in both height and ␴ ͉−1+ 3␣͉ ͑ ͒ Rtc = = ͑24͒ radius Figs. 1 and 2 . Suitable mesh refinement was applied un- ␴c 1+ͱ3␣ der the indenter tip, where large strain gradients occur. The two- dimensional mesh had 12,658 quadratic elements of both types and the axial to transverse yield stress ratio is as follows: CAX6 and CAX8. The characteristic element size in the refined ␴t ␴c ͑q + q ͒2 region was 70 nm ͑see Fig. 2͒. In order to simulate the effect of an A A 0 T ͑ ͒ RAT = = = 25 infinite solid domain in the region far from the indenter tip, linear ␴t ␴c ͑q + q ͒2 T T 0 A infinite elements were used ͓30͔. The three-dimensional finite el- Similar to the elastic formulation, the conditions ement mesh had 57,600 eight-node, hexahedral elements. The el-

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Downloaded 02 Jul 2010 to 18.19.0.100. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 3 Sketch of deformed surface under the indenter. The measure of the sink-in „pile-up… is shown.

Fig. 1 View of the two-dimensional finite element mesh used inelastic phenomena. The method is applied by fitting the top 60% for axial indentation simulations with detail of discretization of the unloading curve using the following fitting equation: under the conical indenter; the left side represents the axis of ͑ ͒n ͑ ͒ symmetry P = A h − hr 27

in which P is the load applied on the indenter, where hr, A, and n are the fitting parameters. The parameter hr has the physical ement characteristic length in the refined region was 30 nm. meaning of residual depth. and indentation hardness H are ءThe analyses were conducted using the commercial finite ele- The indentation modulus E ment code ABAQUS/STANDARD ͓30͔. Due to large displacement and therefore calculated as displacement gradients that occur during an indentation test, large S ␲ ͱ ͑28͒ = ءstrains, and large rotations are expected, therefore the large defor- E mation theory was used in the numerical models. 2 Ap The loading and unloading phases of the experiments are simulated by applying the displacement and repositioning the indenter P H = max ͑29͒ to its original position in two separate steps. The frictional inter- A action between the indenter and bone tissue was modeled by p means of a shear stress proportional to the contact pressure, which respectively, in which S is the contact stiffness dP/dh calculated ͑ ͒ accounts for the Coulomb friction between the surfaces and by by differentiating Eq. 28 at P= Pmax. means of a hard contact algorithm based on Lagrange multipliers S = An͑h − h ͒͑n−1͒ ͑30͒ to account for the contact along the direction normal to the sur- r ͑ ͒ faces. The anisotropic elastoplastic constitutive model here pre- and Ap hp is the projected contact area calculated at maximum sented is implemented into the FORTRAN subroutine UMAT and indentation load Pmax, which is obtained from the finite element 2 linked to the commercial finite element code ABAQUS ͓30͔. solution. For a conical indenter Ap =␲␰ in which ␰ is the radial coordinate of the nodal point in contact with the rigid surface ͑see 2.4 Calculation of Indentation Moduli and Tissue Hard- Fig. 3͒. Calculations for reduced moduli and hardness were car- ness From Numerical Simulations of the Indentation Tests. In ried out for axial as well as transverse indentation simulations. order to compare with experimental data reported in literature, the ͓ ͔ 2.5 Numerical Examples. The parameter sets used in the nu- Oliver–Pharr method 8 was employed to calculate the indenta- merical simulations have been chosen with the purpose to assess ͒ ͑ ͒ء ͑ tion moduli E and hardness H in the axial and transverse the role of anisotropy of yield locus, the deformation modes as- directions from the FEA simulated unloading force ͑P͒ versus sociated to inelastic phenomena ͑nonassociative flow rule͒, and displacement ͑h͒ data. This method is based on the assumption the friction between the indenter and the tissue surface. Further- that the initial part of the P−h unloading curve is unaffected by more, indenter-tissue friction was included in the FEA simulations because of its relevant role in predicting the absence or very limited residual pile-up observed experimentally after nanoindentation ͓17,20͔. The parameters used in the FEA simulations are summarized Table 1. Four sets of analyses have been carried out by changing one parameter at a time ͑see Table 1͒; in particular, the analysis set AT RA refers to simulations of axial indentation in which the ratio

Table 1 Values of the simulation parameters for the four sets of numerical analyses

␣ ␮ ␴c /␴c Analysis set def a T A

AT → RA 0.1 0 0.5 1 AT → Fig. 2 View of the three-dimensional ﬁnite element mesh used RT 0.1 0 0.5 1 → for simulations of transverse indentations; only the detail un- FrA 0.1 0 0.2 0.5 def → der the indenter is shown. Axial direction is the z axis, trans- AA 0.1 0.06 0.2 0.5 verse directions are: r „indentation direction… and c.

081008-4 / Vol. 132, AUGUST 2010 Transactions of the ASME

Downloaded 02 Jul 2010 to 18.19.0.100. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Table 2 Elastic constants employed in the numerical simulations

␮ ͑ ͒ ␭ ͑ ͒ GPa GPa pT pA p0

5.33 13.6 −0.064 0.1286 1

In the last analysis set Adef the nonassociative flow rule for plastic deformation has been used by exploring different values of ␣ ͑ ͒ the dilatation angle def in Eq. 17 . In these analyses, a friction contact between the indenter and the tissue has been assumed ͑␮a =0.2͒. The analysis parameters for each of the above expounded sets are provided in Table 1. The elastic constants were kept un- changed for all simulations; the parameters ␮, ␭, and pi, reported ͑ Fig. 4 Transverse to axial yield stress ratio versus constitutive in Table 2, correspond to the following elastic tensor entries units ͒ parameter qA. Dashed line represents an isotropic yield in GPa : function. C11 = 21.24; C12 = 14.35; C13 = 11.90 ͑ ͒ C22 = 30.91; C44 = 5.63; C55 = 4.67 31 The above values have been obtained by fitting the model param- ␴c /␴c → T A has been chosen in the interval 1 0.5, consistent with eters ␭, ␮, pr, and pa to the elastic constants reported in Ref. ͓15͔ ͓ ͔ ␴c /␴c reported literature 15,28 , where T A =1 represents the case of for human cortical bone. isotropic yield locus. All analyses were carried out in a displacement control mode in The constitutive parameter qA has been selected on the basis of which a full loading-unloading indentation experiment is simu- Eq. ͑25͒ such as a prescribed value of the transverse to axial yield lated. The indenter, initially at contact with the tissue surface, was stress ratio is obtained ͑see Fig. 4͒. The constitutive parameter ¯␴eq displaced downward up to a maximum displacement of 210 nm on ͑␴c loading and thereafter reversed and displaced back to its original was chosen such that axial compressive yield stress is ¯ A = −182 MPa͒ for all analyses, whereas the transverse compressive position on unloading. The constitutive model does not account for time dependent stress changes according to the value of RAT ͑see Fig. 5͒. The constitutive parameter ␣ is set to ␣=0.13, which corresponds to a phenomena, therefore loading rate was not an analysis parameter. fixed tension to compression yield stress ratio Rtc=0.63 and to an ͑␴t ͒ ͓ ͔ axial tensile yield stress ¯ A =115 MPa , as reported in Ref. 15 3 Results ␣ for cortical human femur; the dilatation angle def was set to 0.1. These tensile yield data are consistent with those reported in Refs. The fitting parameters A, hr, and n as well as the maximum ͓31,32͔, which refers to cortical bovine femur data; a slightly indentation force are reported in Table 3 for all simulations. lower value was reported for human cortical femur. AT Similarly, the analysis set RT refers to simulations of transverse indentation tests. All the above analyses are carried out for Table 3 Fitting parameters n, A, and hr „see Eq. „28…… of the ␮a =0. In the analysis set FrA, the friction coefficient between the unloading P−h curves and maximum indentation force „Pmax… indenter and the tissue surface was varied between 0→0.2. for all the performed simulations

A hr Pmax n ͑mN nm−n͒ ͑nm͒ ͑mN͒

AT͑␴c /␴c ͒ RA T A =0.5 1.305 54.840 160.5 1.081 AT͑␴c /␴c ͒ RA T A =0.6 1.341 56.325 155.5 1.130 AT͑␴c /␴c ͒ RA T A =0.7 1.333 53.400 154.5 1.127 AT͑␴c /␴c ͒ RA T A =0.8 1.293 47.316 155.5 1.096 AT͑␴c /␴c ͒ RA T A =0.9 1.251 41.261 157.3 1.039 AT͑␴c /␴c ͒ RA T A =1 1.233 37.932 159.0 0.965 AT͑␴c /␴c ͒ RT T A =0.5 1.144 27.760 175.1 0.596 AT͑␴c /␴c ͒ RT T A =0.6 1.277 37.202 166.2 0.685 AT͑␴c /␴c ͒ RT T A =0.7 1.251 33.154 160.5 0.770 AT͑␴c /␴c ͒ RT T A =0.8 1.252 31.422 155.3 0.826 AT͑␴c /␴c ͒ RT T A =0.9 1.237 28.679 151.3 0.860 AT͑␴c /␴c ͒ RT T A =1 1.229 26.970 147.1 0.899 ͑␮ ͒ FrA a =0 1.305 54.838 160.5 1.081 ͑␮ ͒ FrA a =0.05 1.374 59.382 153.3 1.145 ͑␮ ͒ FrA a =0.1 1.382 56.575 148.9 1.183 ͑␮ ͒ FrA a =0.15 1.356 51.100 147.4 1.191 ͑␮ ͒ FrA a =0.2 1.324 49.967 146.9 1.177 def͑␣ ͒ AA def =0.01 1.326 45.967 146.9 1.178 def͑␣ ͒ Fig. 5 Compressive yield stress along axial and transverse AA def =0.09 1.329 45.899 148.9 1.115 ␴eq def͑␣ ͒ directions versus constitutive parameter qA. The parameter ¯ AA def =0.075 1.323 44.916 153.6 1.037 def͑␣ ͒ has been suitably chosen so that an axial compressive yield AA def =0.060 1.354 48.346 153.6 0.978 stress of 182 MPa is obtained for all values of qA.

Journal of Biomechanical Engineering AUGUST 2010, Vol. 132 / 081008-5

Downloaded 02 Jul 2010 to 18.19.0.100. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 8 Transverse to axial indentation moduli and hardness Fig. 6 Indentation moduli for simulations along axial and ratios versus transverse to axial yield stress ratio transverse indentations versus transverse to axial yield stress ratio

locus, HT /HA =0.98 has been found, indicating that the elastic 3.1 Effect of the Transverse to Axial Yield Stress Ratio anisotropy is not sufﬁcient to justify the direction-dependent in- -dentation hardness observed in experiments. Figure 9 shows con ͒ء ͑ ͒ ء ͑ \ ␴c Õ␴c „ T A =0.5 1…. The axial EA and transverse ET indentation ͑⑀p ͒ ␴c /␴c tour maps of the inelastic shear strain rz under the indenter for moduli for values of T A ranging from 0.5 to 1 are reported in ͑ ␴c /␴c Fig. 6. Consistent with results reported in literature ͓1͔, a higher three reference cases isotropic yield locus, T A =0.8, and ␴c /␴c ͒ ⑀p indentation modulus was obtained in the axial direction ͑20.0– T A =0.5 . From inspecting the contour maps of rz, it can be 22.5 GPa͒ compared with the transverse direction ͑15.0–17.5 observed that for high yield stress transverse to axial mismatch, GPa͒. Limited sensitivity to the yield stress ratio was observed for the inelastic shear strain magnitude and its spatial gradient along ͑ ⑀p both the directions. the radial direction increase see also Fig. 10 for line plots of rz The axial and transverse hardness as a function of the ratio as a function of the distance from the indentation point͒. This ␴c /␴c leads to a higher contact depth with respect to the isotropic case, T A are reported in Fig. 7. Similar to the indentation moduli, the axial hardness values were also found to be higher in the axial resulting in an increase in the contact area, which in turn de- direction compared with the transverse indentation simulations. creases the hardness. It must be kept in mind that, in this para- However, hardness exhibited a high sensitivity to the transverse to metric study, the axial yield stress is kept constant; whereas, the axial yield stress ratio. In particular, axial hardness was found to transverse yield stress is not; therefore transverse hardness is con- c c ␴c /␴c ␴ /␴ ͑ tinuously decreasing with T A consistently with the decrease in range from about 0.90 GPa for T A =1 i.e., isotropic yield c ͒ ␴c /␴c ␴ . For comparison purposes, Fig. 8 reports also the transverse to locus to 0.72 GPa for T A =0.5; whereas, transverse hardness T ␴c /␴c axial indentation moduli ratio; it can be observed that the inden- was found to range from 0.90 GPa for T A =1 to 0.51 GPa for ␴c /␴c tation moduli ratio is largely unaffected by the yield stress ratio. T A =0.5. Furthermore, the transverse to axial hardness ratio / ␴c /␴c ͑ The indentation simulations allowed the calculation of the HT HA is also highly sensitive to the parameter T A see Fig. pile-up at a maximum penetration depth of 210 nm ͑Fig. 11͒. ͒ / ␴c /␴c 8 . The solid line in Fig. 8 represents the case of HT HA = T A. According to the sketch provided in Fig. 3, the pile-up here is ␴c /␴c Ͻ The results show that for T A 0.8 the hardness ratio is higher measured as the vertical displacement of the external boundary of than the yield stress ratio ͑an ampliﬁcation effect͒; whereas, for the contact area with respect to the original surface level. Positive ␴c /␴c Ͼ / ␴c /␴c T A 0.8, HT HA = T A. In particular, for an isotropic yield values of pile-up indicate that the material rises with respect to the undeformed surface ͑typical for ductile materials͒; whereas, negative values indicate that the material sinks-in with respect to the undeformed surface. This latter case is typical of ceramic brittle materials with a low elasticity to yield stress ratio. The set of AT analyses RA shows an increasing pile-up for decreasing transverse to axial yield stress ratio. A maximum pile-up of 40 nm has ␴c /␴c ͑ ␮ ͒ been found for T A =0.5 Fig. 11 curve labeled a =0 .

3.2 Effects of the Inelastic Deformation Mode and of Friction. The effect of friction between indenter and tissue surface was investigated in the analyses set FrA ͑see Table 1͒ in which axial indentations were simulated. It was determined that friction affects both the pile-up ͑see Fig. 11͒ and the indentation hardness ͑see Fig. 12͒. The increase in friction coefﬁcient substan- ␴c /␴c ͑ ͒ tially decreases the pile-up for all values of T A Fig. 11 . For ␮ Ͼ ␴c /␴c Ͼ friction coefﬁcients a 0.1 and for T A 0.75, sink-in is observed. Simultaneously, increases in tissue hardness are observed up to ␮a =0.2 ͑see Fig. 12͒; for larger values of the ␮a no further hardness increases take place ͑data not shown͒. Notably, friction ͑ ͒ Fig. 7 Indentation hardness for simulations along axial and does not affect the axial indentation modulus see Fig. 12 . transverse indentations versus transverse to axial yield stress Furthermore, the contribution of the inelastic deformation mode ␴c /␴c ratio is required to avoid piling-up of the material for T A =0.5. The

081008-6 / Vol. 132, AUGUST 2010 Transactions of the ASME

Downloaded 02 Jul 2010 to 18.19.0.100. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm ⑀p Fig. 9 Contour maps of the inelastic strain rz component „ rz, where z and r are the directions parallel and perpendicular to the long bone axis direc- ␴c ␴c tion, respectively… in an axial indentation simulation for: T / A =1 „left…, ␴c ␴c ␴c ␴c T / A =0.8 „center…, and T / A =0.5 „right…

def set of analyses AA have shown that the inelastic deformation 4 Discussions and Conclusions ␣ mode, governed by the parameter def, has a moderate effect on the axial hardness and indentation moduli ͑Fig. 13͒; whereas, it 4.1 Origins of Direction-Dependent Response. The trans- strongly affects the pile-up ͑Fig. 14͒. Indeed, as shown in Fig. 14, versely isotropic elasticity assumption is a primary determinant of ͑ the pile-up decreases markedly with decreasing ␣ . For ␣ the observed indentation moduli anisotropy ratio Fig. 8 compared def def ͑ ͒͒ Ͻ0.09, sink-in effect is observed. In this case, both direction- with Eq. 31 . It was conﬁrmed that the degree of plastic aniso- ͑ ͒ tropy ͑including the isotropic yield locus͒ had little effect on the dependent hardness Fig. 8 and absence of piling-up of the ma- ͑ ͒ terial ͑Fig. 14͒ are simultaneously observed. direction-dependent elastic moduli Fig. 6 , which is attributed to the fact that little plasticity takes place during the initial stage of the unloading P−h curve. The slight dependence of the indentation moduli on the transverse to axial yield stress ratio is attributed to differences in the computed contact area at maximum load. It was also shown that anisotropic post-elastic properties, i.e., anisotropic yield locus and plastic potential, are required in order

⑀p Fig. 10 Plots of the inelastic strain rz component as a function of the distance from the indentation point on the surface c c c c Fig. 12 Axial hardness versus friction coefﬁcient „left vertical for: isotropic yield locus model, ␴ /␴ =0.8, and ␴ /␴ =0.5 ␣ T A T A axis… for def =0.1; axial hardness versus friction coefﬁcient ␣ „right vertical axis… for def =0.1

␣ Fig. 11 Pile-up versus transverse to axial yield stress ratio for Fig. 13 Axial hardness „left vertical axis… versus def; axial in- ␣ three different values of tissue-probe friction coefﬁcient dentation modulus „right vertical axis… versus def

Journal of Biomechanical Engineering AUGUST 2010, Vol. 132 / 081008-7

Downloaded 02 Jul 2010 to 18.19.0.100. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 15 Indentation modulus and indentation hardness from FEM simulations in the axial direction „symbols…; shaded area represent the values „mean±sd… from experiments †1‡ ␣ ␮ Fig. 14 Axial indentations: pile-up versus def for a =0.2

Ϯ ء ͔ ͓ to obtain a direction-dependent tissue hardness ͑see Fig. 7͒. In- experiments in Ref. 1 reported ET =16.6 1.1 GPa and HT deed, isotropic yield surface predicts the values of the ratio of =0.56Ϯ0.034 GPa, respectively. Therefore, good agreement was transverse to axial hardness very close the yield anisotropy ratio obtained for indentation moduli as well as for indentation hard- ͑Fig. 8͒. However, interestingly an amplification effect is ob- ness between predicted and experimentally reported values. served, whereby hardness anisotropy ratios exceed yield stress Another key result of this study is that pile-up of the material is ͑ ͒ ͑ ͒ anisotropy ratios for ␴T /␴A Ͻ0.8. This effect is owed to the de- affected by: i the tissue-indenter friction and ii the inelastic creasing of transverse yield stress, which produces a decreasing deformation mode ͑in this model represented by the dilatation ␣ ͒ axial hardness for ␴T /␴A Ͻ0.8. angle def . In particular, the pile-up is small or even becomes negative ͑i.e., sink-in͒ when the dilatation angle is decreased and 4.2 Comparison With Previously Reported Experimental when the probe-tissue friction is accounted for. This finding has a Data. The results are highly consistent with experimental data of great relevance since experiments have shown that bone tissue literature ͓1,4–6͔. The indentation moduli values reported in Fig. indentation produce a negligible pile-up ͓17,20͔. High values of 6 are comparable to those obtained by Rho et al. ͓1͔͑22.4 GPa ␣ ͑ ␣ ␣ the dilatation angle def with the upper limit def= , i.e., asso- and 16.6 GPa in the axial and transverse directions, respectively͒, ciative flow rule͒ implies high volumetric dilatation for all values ͓4͔͑22.5 GPa, axial direction͒, and ͓5͔͑21.7 GPa, axial direction͒ of the hydrostatic stress component. It is therefore justifiable to for human tibial cortical tissue. Regarding hardness ͑Fig. 7͒, ͓1͔ ␣ assume small values of def to reduce the volume increase during reports a transverse to axial hardness ratio of 0.91 for human tibial the inelastic deformation process. cortical tissue. This anisotropy ratio corresponds to the one obtained in this study for a transverse to axial yield stress ratio 4.3 Model Limitations. In the following we provide a sum- ␴c /␴c ͑ ͒ mary of the limitations of the current model. T A =0.9 Fig. 8 . For bovine bone, the transverse to axial hard- The elastic-plastic constitutive law does not represent the most ness ratio HT /HA =0.76 reported in Ref. ͓6͔ as well as the magnitude of the hardness values H =811 MPa and H =647 MPa are general anisotropic transversely isotropic stress-strain relation- A T ͑ ͒ ␴c /␴c ship. In particular, the elastic tensor 6 in which r2 =r3 =rT is all consistent with a transverse to axial yield stress ratio T A Ϸ0.6 in this model. Compressive and tensile axial yield stresses assumed, depends on three independent parameters instead of five, ͑ ͒ as required by transverse isotropy; this issue is discussed in Ref. used in the above analyses 180 MPa and 115 MPa, respectively ͓ ͔ correspond to a material cohesion ¯␴eq=141 MPa, as defined in 24 . Furthermore, the post-yield constitutive law is characterized the Drucker–Prager strength criterion ͑14͒. This material param- by a compressive to tensile yield stress ratio independent of material direction; which does not match exactly with experimental eter is consistent with that found in Ref. ͓17͔͑¯␴eq=122 MPa͒, finding ͑see, for example, Ref. ͓15͔͒. This model has the advan- which refers to microscale indentation experiments and it is lower tage to be dependent on fewer parameters, as opposite to more ͓ ͔͑␴eq ͒ than that found in Ref. 16 ¯ =171 MPa , which refers to mac- general models, and simpler sensitivity analysis. The results have roscale experiments. These differences may suggest that a size shown that the above mentioned limitations do not affect the main dependence exists with the cohesive strength increasing with in- conclusions described in this research. creasing size scale as expected when investigating tissues exhib- A further limitation concerns the assumed conical geometry for iting a hierarchical structure. the indenter. This is a typical assumption for finite element simu- In order to carry out a quantitative comparison of the FEA -lations of nanoindentation and the equivalence between simula ء predictions to the magnitude of E and H reported in Ref. ͓1͔, tions carried out with spherical indenters and those with actual axial indentation simulations were run to different penetration pyramidal geometry has been well documented ͓34,35͔. ͑ ͒ ͑ depths hmax up to 500 nm which is consistent with that used in It should be also underlined that the effect of pore water has Ref. ͓1͔͒. Since the FEA simulations assume a homogeneous con- been disregarded in this study. The water acts as a plasticizer for tinuum ͑rather than taking into account specific microstructural the collagenous phase filling the 50% volume fraction of inter- features, as in Ref. ͓33͔͒, the estimated mechanical properties, fibrillar and the 20% of the extrafibrillar space and may have a such as indentation modulus and hardness, are, by definition, in- significant effect on tissue anisotropy properties, as shown in sensitive with respect to length scale ͑maximum force and depth͒. Refs. ͓36–38͔, in which hydrated and dehydrated bone tissues, Figure 15 verified that the axial indentation modulus and hardness where subject to nanoindentation. In the above mentioned papers, were indeed approximately independent from penetration depth; the effect of hydration on the transverse to axial ratio of the in- in the same figure the shaded area bounds the experimental find- dentation response was investigated by means of spherical inden- ء ings ͑meanϮsd͒ for both E and H from Ref. ͓1͔. In the trans- tation with tip radius of 20 ␮m. In particular, in Ref. ͓36͔,itwas verse direction, the computed indentation modulus and hardness found that hydrated bone samples did not show a statistically sig- ء were ET =16.39 GPa and HT =0.58 GPa, respectively; whereas, nificant transverse to axial indentation moduli ratio, in contrast to

081008-8 / Vol. 132, AUGUST 2010 Transactions of the ASME

Downloaded 02 Jul 2010 to 18.19.0.100. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm ϭ indentation modulus ءwhat was found on dehydrated samples. The model presented in E ϭ ء ͔ ͓ this paper applies for dehydrated tissue samples as in Refs. 1,39 ; EA indentation modulus along axial direction ϭ ء however, it is still to be investigated whether the independence of ET indentation modulus along transverse direction indentation moduli on test directions found for hydrated samples eij ϭ deviatoric strain components in Ref. ͓36͔ is owed to the specific indenter geometry. Further i ϭ gk ;i,k=1..3 components of unit vectors along principal studies are needed in order to extend these findings to pyramidal material directions indentations. H ϭ hardness The choice of a plasticity framework for the numerical simula- HA ϭ hardness along axial direction tion of indentation of the bone tissue is widely accepted ͑one of ϭ ͓ ͔͒ HT hardness along transverse direction the most recent papers which deals with this issue is Ref. 17 . h ϭ residual indentation depth Although it is well known that plastic deformation, as defined for r Iijkl ϭ fourth-order unit tensor metals, are not physical in bone tissue, it has been shown by 2 ϭ several experimental studies on the nanoscale deformation modes Mij second-order constitutive tensor 4 ϭ of bone that a stress-strain curve of tissue exhibits a nonlinear Mijkl fourth-order constitutive tensor ϭ relationship with the achievement of a “yield” stress, which may n parameter for fitting P−h curve ϭ be correlated with the onset progressive disruption of the sacrifi- P indentation load ϭ cial bonds during which the tangent to the stress-strain curve pro- Pmax maximum indentation load gressively decreases ͓40–43͔. This behavior can be phenomeno- pi ; i=0.3 or logically reproduced by the elastic-plastic constitutive law. The i=A,T ϭ weight parameters for elastic anisotropy ء disruption of sacrificial bonds, however, generates a decrease in p ϭ hydrostatic pressure in fictitious anisotropic elastic modulus of the tissue in a damage-like mechanism. An space appropriate physically consistent post-yield model should define a qi ; i=0.3 or flow rule, i.e. the law ruling the development of inelastic strain, i=A,T ϭ weight parameters for inelastic anisotropy accounting for deformation energy, which has been shown to be RAT ϭ axial/transverse yield strength ratio dissipated by shearing of a thin “glue” layer between mineral- Rtc ϭ tension/compression yield strength ratio reinforced collagen fibrils ͓43͔. Further developments of the pre- S ϭ slope of P−h curve ϭ ء sented model accounting for the most relevant above specified Sij deviatoric stress components in fictitious aniso- limitations will improve our knowledge of the post-elastic behav- tropic space ior of the cortical bone tissue and its relationship with the tissue ␣ ϭ Drucker–Prager constitutive parameter ␣ ϭ constitution. def dilatation angle In summary, the technique of indentation is increasingly being ␦ij ϭ Kronecker delta function used for bone mechanics characterization and has potential for use ⑀ij ϭ total strain components as a diagnostic tool. Our study provides a rigorous scientific link- ⑀p ϭ rz inelastic strain components age between refined constitutive models representing the tissue’s ␭ϭelastic constitutive parameter fundamental material properties and the complex multiaxial stress ␮ ϭ elastic constitutive parameter field generated during a penetration. This work provides interest- ␮a ϭ friction coefficient ing results on the relationship between bone indentation aniso- ␴c ϭ A compressive strength along the axial direction tropy and anisotropy in uniaxial stress fields. More specifically, ␴t ϭ the model presented is able to predict direction-dependent inden- A tensile strength along the axial direction ␴c ϭ tation moduli, direction-dependent hardness, and negligible T compressive strength along the transverse pile-up simultaneously. In the future, this model may be used to direction ␴t ϭ assess the effect of variations of constitutive parameters due to T tensile strength along the transverse direction ␴ ϭ age, injury, and/or disease on bone mechanical performance in ij components of Cauchy stress tensor multiaxial stress fields. Furthermore, microstructurally based FEA ¯␴eq ϭ material cohesion parameter ϭ ء␴ models could be coupled to the present continuum-based approach ij stress components in fictitious anisotropic to assess the role of localized deformation mechanisms and spe- space -ϭ Von Mises equivalent stress in fictitious aniso ءcific microstructural features ͓40–42͔ and to determine the limita- ¯␴ tions and applicability of the continuum approach. tropic space ␾ ϭ Drucker–Prager yield locus Acknowledgment The authors would like to thank the MIT-Italy Program and the References Progetto Rocca for supporting Davide Carnelli’s one year schol- ͓1͔ Rho, J., Roy, M., Tsui, T., and Pharr, G., 1999, “Elastic Properties of Micro- arship at the MIT. We also gratefully acknowledge the U.S. Army structural Components of Human Bone Tissue as Measured by Nanoindenta- through the MIT Institute for Soldier Nanotechnologies ͑Contract tion,” J. Biomed. Mater. Res., 45, pp. 48–54. ͓ ͔ No. DAAD-19-02-D0002͒, the National Security Science and En- 2 Ebenstein, D., and Pruitt, L., 2006, “Nanoindentation of Biological Materials,” ͑ ͒ Nanotoday, 1, pp. 26–33. gineering Faculty Fellowship NSSEFF Program, the National ͓3͔ Lewis, G., and Nyman, J., 2008, “Review, The Use of Nanoindentation for Science Foundation MIT Center for Materials Science and Engi- Characterizing the Properties of Mineralized Hard Tissues: State-of-the Art neering ͑Contract No. DMR-0819762͒, and the NSF under Grant Review,” J. Biomed. Mater. Res., Part B: Appl. Biomater., 87B, pp. 286–301. ͓ ͔ No. CMMI-0758651. Furthermore, Riccardo Lucchini and Matteo 4 Rho, J., Tsui, T. Y., and Pharr, G. M., 1997, “Elastic properties of Human Cortical and Trabecular Lamellar Bone Measured by Nanoindentation,” Bio- Ponzoni are gratefully acknowledged for their contribution in per- materials, 18, pp. 1325–1330. forming the numerical simulations. ͓5͔ Rho, J., Zioupos, P., Currey, J. 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